Question

Quadratic Functions in Context(1 point)
A hawk swoops down from its perch 30 feet in a tree to catch field mice. It follows the function h(t)=6t2−24t+30
, where t
is time in seconds and h
is height in feet.

Part A: Graph this function. Link to Desmos calculator

Part B: What is the initial height of the hawk? Write your answer as an ordered pair

Part C: At which point is the hawk at it's lowest height? Write your answer as an ordered pair

Part D: The hawk will be at it's lowest point at time
seconds at a height of
feet

Answer 1

Let's break down each part of your question regarding the quadratic function \( h(t) = 6t^2 - 24t + 30 \).

### Part A: Graph this function
You can use the Desmos calculator to input the function. You will need to go to [Desmos Graphing Calculator](https://www.desmos.com/calculator) and enter the function \( h(t) = 6t^2 - 24t + 30 \). This will give you the graph of the hawk's height over time.

### Part B: What is the initial height of the hawk?
The initial height of the hawk occurs when \( t = 0 \). To find this, substitute \( t = 0 \) into the function:

\[
h(0) = 6(0)^2 - 24(0) + 30 = 30
\]

So the initial height is \( 30 \) feet. As an ordered pair, this is:
\[
(0, 30)
\]

### Part C: At which point is the hawk at its lowest height?
The lowest point of a quadratic function (a parabola that opens upwards) can be found using the vertex formula. The time at which the vertex occurs is given by:

\[
t = -\frac{b}{2a}
\]

where \( a = 6 \) and \( b = -24 \).

Calculating:
\[
t = -\frac{-24}{2 \cdot 6} = \frac{24}{12} = 2
\]

Now, plug \( t = 2 \) back into the height function to find the height:

\[
h(2) = 6(2)^2 - 24(2) + 30 = 6 \cdot 4 - 48 + 30 = 24 - 48 + 30 = 6
\]

Thus, the lowest height occurs at:
\[
(2, 6)
\]

### Part D: The hawk will be at its lowest point at time ___ seconds at a height of ___ feet.
From our calculations, the hawk will be at its lowest point at:
- Time: \( 2 \) seconds
- Height: \( 6 \) feet

So, you can summarize your answer as:
The hawk will be at its lowest point at time **2** seconds at a height of **6** feet.